In a few of previous works of our group
[1]}, [2]} - [3]} we illustrated how such complex behavior might appear in the frame work of the simple Zwanzig
model, and how the simplest version of the model can be generalized to relax some unphysical assumptions of the model. Besides in these works we proposed a method which makes it possible to solve the dynamical problem analytically beyond the bare Zwanzig model approximations (
[4]} - [3]}).
The main ingredients of our new method (only schematically and briefly described in the previous papers) is the representation of the partial amplitudes of recurrence cycles. Unlike the standard Fourier expansion over eigen-frequencies, this representation reveals explicitly the time dependent exchange between intra-molecular states in each recurrence cycle. The fine structure of the Loschmidt echo arises as a result of the dephasing phenomena associated with the fact that the exchange of the different reservoir states with the initial state occurs not at the same instants of time. The synchronization of the reverse transitions is destroyed when the Loschmidt echo components of the different recurrence cycles start to overlap. The cycle overlapping determines the critical recurrence cycle number. Then at larger time (cycle numbers) the system dynamics is expected to be in the stochastic-like regime.
